E (mathematical constant)

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The correct title of this article is e (mathematical constant). The initial letter is capitalized due to technical restrictions.
Image:E-ruud.png
e is the unique number, such that the derivative of ex at x=0 equals 1.

The mathematical constant e is the base of the natural logarithm. It is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions; some of them are given below. Its value to the 29th decimal digit is:

e ≈ 2.71828 18284 59045 23536 02874 7135

Contents

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The first indication of e as a constant was discovered by Jacob Bernoulli, trying to find the value of the following expression.

\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel after a, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his first initial, since he was a very modest man, always trying to give proper credit to the work of others.1

Definitions

The three most common definitions of e are listed below.

  1. The limit
    e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n
  2. The sum of the infinite series
    e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots
    where n! is the factorial of n.
  3. The unique real number x > 0 such that
    \int_{1}^{x} \frac{1}{t} \, dt = {1}
    (that is, the number x such that area under the hyperbola f(t) = 1 / t from 1 to x is equal to 1).

These different definitions can be proven to be equivalent.

Properties

The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive:

\frac{d}{dx}e^x=e^x and
\int e^x\,dx=e^x + C, where C is the arbitrary constant of integration.

It is known that e is irrational (proof) and even more, transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula, one of the most important formulas in mathematics:

e^{ix} = \cos(x) + i\sin(x),\,\!

described by Richard Feynman (p. I-22-10) as "Euler's jewel".

The special case with x = π is known as Euler's identity:

e^{i\pi}+1 =0 .\,\!

The following is an infinite simple continued fraction expansion of e (sequence A005131 in OEIS):

e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, \ldots,1, 2n, 1,\ldots] \,

The following is an infinite generalized continued fraction expansion of e:

e= 2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{\ddots}}}}.

The number e is also equal to the sum of the following infinite series:

e = \left [ \sum_{k=0}^\infty \frac{(-1)^k}{k!} \right ]^{-1}
e = \left [ \sum_{k=0}^\infty \frac{1-2k}{(2k)!} \right ]^{-1}
e = \frac{1}{2} \sum_{k=0}^\infty \frac{k+1}{k!}
e = 2 \sum_{k=0}^\infty \frac{k+1}{(2k+1)!}
e = \sum_{k=0}^\infty \frac{3-4k^2}{(2k+1)!}
e = \sum_{k=0}^\infty \frac{3k^2+1}{(3k)!}
e = \left [ \sum_{k=0}^\infty \frac{4k+3}{2^{2k+1}\,(2k+1)!} \right ]^2
e = \frac{-12}{\pi^2} \left [ \sum_{k=1}^\infty \frac{1}{k^2} \ \cos \left ( \frac{9}{k\pi+\sqrt{k^2\pi^2-9}} \right ) \right ]^{-1/3}
e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}

The number e is also given by several infinite product forms including the Pippenger product

e= 2 \left ( \frac{2}{1} \right )^{1/2} \left ( \frac{2}{3}\; \frac{4}{3} \right )^{1/4} \left ( \frac{4}{5}\; \frac{6}{5}\; \frac{6}{7}\; \frac{8}{7} \right )^{1/8} \cdots

as well as,

\frac{2\cdot 2^{(\ln(2)-1)^2} \cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^3}\cdots }

The number e is equal to the limit of several infinite sequences:

e= \lim_{n \to \infty} n\cdot\left ( \frac{\sqrt{2 \pi n}}{n!} \right )^{1/n} and
e=\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} (both by Stirling's formula).

The symmetric limit,

e=\lim_{n \to \infty} \left [ \frac{(n+1)^{n+1}}{n^n}- \frac{n^n}{(n-1)^{n-1}} \right ]

may be obtained by manipulation of the basic limit definition of e. Another limit is

e= \lim_{n \to \infty}(p_n \#)^{1/p_n}

where pn is the nth prime and p_n \# is the primorial of the nth prime.

It was shown by Euler that the infinite tetration

x^{x^{\cdot^{\cdot^{\cdot}}}},

converges only if e^{-e} \le x \le e^{1/e}.

Non-mathematical uses of e

One of the most famous mathematical constants, e is also frequently referenced outside of mathematics. Some examples are:

  • In the IPO filing for Google, in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is, of course, e billion dollars to the nearest dollar.
  • Google was also responsible for a mysterious billboard [1] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts, which read {first 10-digit prime found in consecutive digits of e}.com. Solving this problem and visiting the web site advertised led to an even more difficult problem to solve. (The first 10-digit prime in e is 7427466391, which surprisingly starts as late as at the 101st digit.) [2]


References

  • Maor, Eli; e: The Story of a Number, ISBN 0691058547
  • O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: "The number e"; University of St Andrews Scotland (2001)

Notes

Template:Ent O'Connor, "The number e"

External links

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